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3 Statistics on Articulated Models
An articulated model is well adapted to represent the spine because it intuitively
describes its natural degrees of freedom. Building statistical models based on this
type of articulated representation is therefore very attractive. However, there are
theoretical complications. Because articulated models rely on rigid transformations
to encode inter-vertebral transformations, it is necessary to compute the statistics on
these rigid transformations.
Rigid transformations are special because they cannot be added together like real
numbers. Unfortunately, concepts as simple as the mean or standard deviation
require summing the measurements as part of their computation. This calls for the
generalization of a few basic concepts.
These generalizations are performed using a few mathematical tools borrowed
from the
field of Riemannian geometry. These tools will be introduced as simply as
possible; thus, no prior knowledge of Riemannian geometry is needed. Neverthe-
less, interested readers can
find a more complete introduction in [
6
].
first discuss a few properties of the rigid transformations related to
Riemannian geometry that will be needed to build a statistical model of the spine.
Then, we will present the generalization of the mean and covariance that are used in
articulated statistical models of the spine. Finally, the visualization of the mean and
covariance of the articulated spine models will be discussed.
We will
3.1 Riemannian Geometry and Rigid Transformations
The most common method for numerically representing a rigid transformation T is
to use the combination of a rotation matrix R and translation vector tT
.
Using this representation, the action of T on a 3D point x can be written as
y
ð
¼
fg
R
;
t
Þ
¼
Rx
þ
t, and the composition of two rigid transformations T
2
and T
1
is given by
T
2
.
The composition and action on points have very simple and ef
T
1
¼
f
R
2
R
1
;
R
2
t
1
þ
t
2
g
cient expressions
using this representation. Thus, it may be tempting to compute statistics directly on
R and t. However, naively summing rotation matrices and dividing the number of
transformations in an attempt to compute an average rotation matrix will most likely
result in a matrix that is not a rotation matrix. It could even lead to a singular matrix.
Fortunately, there are several other ways to represent rotations beyond the con-
ventional rotation matrix. For instance, Euler angles are a compact notation that can
be useful in certain applications. Unit quaternions require less mathematical opera-
tions to perform multiple compositions. Unfortunately, computing statistics directly
on these representations leads to problems because the results depend on the orien-
tation of the global frame of reference. Another, perhaps less known representation,
called the rotation vector, offers certain advantages. This representation is de
ned
using an axis of rotation n and an angle of rotation
(see Fig.
3
). The rotation vector
θ
